A301590 - OEIS

A301590

Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 257, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389

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OFFSET

1,1

COMMENTS

In the definition, A023900(p) = 1-p. One has sign(A023900(n)) = (-1)^A001221(n), so a different solution x can only exist if x has at least 3 distinct prime factors. The smallest number of the form p*q*r such that (p-1)*(q-1)*(r-1) = P-1 for primes p, q, r, P is 2*3*7 = 42, eliminating P = 13 = A301591(1) from this sequence. This is the case whenever (P+1)/2 = p > 3 is a prime (in A005382), whence P-1 = (2-1)*(3-1)*(p-1), which eliminates all P > 5 in A005383 from this sequence. - M. F. Hasler, Aug 14 2021

LINKS

M. F. Hasler, Table of n, a(n) for n = 1..10000, Sep 01 2021

EXAMPLE

2 is a term because there are no other solutions to A023900(x) = A023900(2) = -1 than other powers of 2.

13 is not a term because A023900(42) = -12 = A023900(13). Similarly, no P > 5 in A005383 is a term because A023900(P) = 1-P = (1-2)*(1-3)*(1-p) = A023900(2*3*p) with p = (P+1)/2. - M. F. Hasler, Aug 14 2021

PROG

(PARI) f(n) = sumdivmult(n, d, d*moebius(d)); /* A023900 */

isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0)); ); return (1); }

lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(p, ", ")); ); }

(PARI) select( {is_A301590(p)=!forcomposite(k=p+1, p^2-1, A023900(k)!=1-p|| return)&& isprime(p)}, primes([1, 399])) \\ M. F. Hasler, Aug 14 2021

CROSSREFS

Cf. A023900, A301374.

Complement (within the primes) of A301591, which has A005383 \ {3, 5} as a subsequence. Appears to have A079151 \ {13} as subsequence.

Sequence in context: A042988 A167135 A129990 * A293194 A162566 A040085

Adjacent sequences: A301587 A301588 A301589 * A301591 A301592 A301593

KEYWORD

nonn

AUTHOR

Michel Marcus, Mar 24 2018

STATUS

approved